Have ridden BRP many times and the only traffic problems I've encountered are in the Fall when the "leaf watchers" show up and drive 15 mph. Traffic will be lighter during the week, so if you can avoid the weekend, do so. I've ridden the Parkway from Asheville to the VA line and back in a day and made acceptable time. You'll make better time the further north you go due to the flatter terrain. If your schedule allows, try to work in a trip on the Cherohala Skyway. It's a BRP type drive, but not nearly as long. Also, I've been to the BRP in June, and at the higher elevations in the a.m., had on long-johns, chaps, sweatshirt, leather jacket, winter gloves, etc. Enjoy your trip.